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High-pressure strength and compressibility of titanium diboride (TiB2) studied under non-hydrostatic compression
Journal of Physics and Chemistry of Solids 121 (2018) 256–260
Contents lists available at ScienceDirect
Journal of Physics and Chemistry of Solids journal homepage: www.elsevier.com/locate/jpcs
High-pressure strength and compressibility of titanium diboride (TiB2) studied under non-hydrostatic compression
T
Hao Lianga,b, Haihua Chena,∗, Fang Pengb,∗∗, Lingxiao Liua, Xin Lib, Kui Liua, Chuangqi Liuc, Xiaodong Lid a
Foundation Department of Qinghai University, Xining, 810016, PR China Institute of Atomic and Molecular Physics, Sichuan University, Chengdu, 610065, PR China c College of Optoelectronic Technology, Chengdu University of Information Technology, Chengdu, 610225, PR China d Institute of High Energy Physics, Chinese Academy of Sciences, Beijing, 100049, PR China b
A R T I C LE I N FO
A B S T R A C T
Keywords: Titanium diboride Non-hydrostatic pressure Bulk modulus Yield strength
The high-pressure strength and plastic properties of titanium diboride (TiB2) were investigated using synchrotron angle-dispersive x-ray diffraction (AXRD) under non-hydrostatic compression up to 42 GPa in a diamondanvil cell (DAC). The AXRD data yielded a bulk modulus K0 = 308 ± 10 GPa with a pressure derivative K′0 = 3.4 ± 1. The experimental data are discussed and compared to the results of first-principles calculations. The compressibility of TiB2 demonstrates a strongly anisotropic property with increasing pressure. In addition, the microscopic deviatoric stress and grain size (crystallite size) were determined as a function of pressure from the line-width analysis. We can seen that the strength increases while the crystalline size decreases steeply as the pressure is raised from ambient to about 22 GPa. In other words, TiB2 starts to yield a plastic deformation at around 22 GPa, and the yield strength of TiB2 increases with pressure, reaching a value of ∼27 GPa at the highest pressure in our experiment.
1. Introduction Transition metal diboride ceramics (TiB2, CrB2, TaB2, HfB2) are actively investigated because of their useful physical and mechanical characteristics [1]. Titanium diboride (TiB2) materials have received wide attention because of their high melting points (about 3000 °C), high hardness (Vickers hardness = 33.7 GPa and Mohs hardness > 9) and elastic modulus, high resistance to corrosion and wear, and fair chemical inertness in metal melts and excellent thermal shock resistance [2,3]. Due to their outstanding properties, they have been extensively used for high-temperature structural materials, cutting tools and hard coating, and have aroused great interest in basic research and several technological applications [4]. It has been reported that TiB2 has a hexagonal lattice structure (C32) of space group no. 191 (P6/ mmm), and the lattice parameters were a = 3.0292 Å and c = 3.2284 Å. There are three atoms in the unit cell, all of them in special positions: the Hf atom is on the 1a Wyckoff site (0, 0, 0) and the B atoms are on the 2 d Wyckoff site (1/3, 2/3, 1/2) [5]. A hexagonal TiB2 crystal has six different elastic coefficients (C11, C12, C13, C33, C44 and C66), but only five of them are independent since C66 = 1/2 (C11 –
∗
C12) [6]. In previous studies [7–15], the bulk modulus of titanium diboride, which ranges from 193 GPa to 399 GPa, has been intensively investigated using experiments and calculations. Several studies have reported on the bonding mechanism, lattice parameters, and elastic properties of TiB2 using first-principles local-density-functional calculations [16]. Knowledge of the elastic properties, strength, and plastic deformation of TiB2 ceramic materials is a requisite for understanding its stability and for its applications under extreme static and dynamic mechanical stresses and high temperature. In 2006, TiB2 was studied by Amulele et al. [17] in a DAC in order to determine its hydrostatic equation of state (EOS) and investigate its strength. The measurements were conducted up to 60 GPa by using the radial x-ray diffraction technique and analyzed using the lattice strain theory [18–20]. Unfortunately, there are relatively few studies devoted to direct experimental measurements on the strength and plastic deformation behavior of titanium diboride by analyzing the x-ray diffraction peak broadening under non-hydrostatic compression. In this study, the EOS, high-pressure strength and plastic properties of a polycrystalline titanium diboride sample were investigated using a diamond anvil cell (DAC) by synchrotron angle-dispersive x-ray
Corresponding author. Foundation Department of Qinghai University, Xining, 810016, PR China. Corresponding author. Institute of Atomic and Molecular Physics, Sichuan University, Chengdu, 610065, PR China. E-mail addresses: [email protected] (H. Chen), [email protected] (F. Peng).
∗∗
https://doi.org/10.1016/j.jpcs.2018.05.042 Received 15 December 2017; Received in revised form 25 May 2018; Accepted 26 May 2018 Available online 26 May 2018 0022-3697/ © 2018 Published by Elsevier Ltd.
Journal of Physics and Chemistry of Solids 121 (2018) 256–260
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Fig. 1. (a) X-ray diffraction patterns (λ = 1.5404 Å) of the initial powder under normal pressure and room temperature. Upper-right inset: scanning electron microscope (SEM) images of starting powder. (b) Schematic crystal structures of TiB2.
9 × 9 × 2 Monkhorst–Pack k-point sampling was used for all cases. The tolerances for geometry optimization were set as the difference of total energy within 5 × 10−6 5 eV/atom, maximum ionic Hellmann-Feynman force within 0.01 eV/Å, maximum ionic displacement within 5 × 10−4 Å, and maximum stress within 0.02 GPa. Convergence tests revealed that the parameters described above were sufficient to lead to a well-converged total energy, which are carefully tested in the text. The bulk modulus at ambient pressure (K0) and its pressure derivative (K 0′) were determined by fitting the numerical data to the isothermal Birch-Murnaghan EOS.
diffraction (AXRD) under uniaxial compression up to 42 GPa. For comparison with previous and experimental results, the elastic properties of TiB2 were calculated using first-principles density functional theory (DFT). 2. Material and methods 2.1. Experimental procedures Polycrystalline titanium diboride powder (99.99% purity, 0.5–1 μm grain size) was purchased from the Alfa Aesar Co., Ltd., Shanghai China. The initial powder was characterized by x-ray diffraction (XRD; model DX-2500, Dandong, China) using a Cu Kα radiation source with λ = 1.5404 Å to check the crystal structure, purity and morphology. Fig. 1a shows the power XRD pattern and scanning electron microscope (SEM) images of the starting specimen. As shown in Fig. 1b, TiB2 has a hexagonal structure. An in situ high pressure AXRD experiment was carried out using a symmetric-type DAC with a culet size of 300 μm, which was interfaced with angle-dispersive synchrotron radiation at the 4W2 beamline of the Beijing Synchrotron Radiation Facility (BSRF, China) at room temperature. TiB2 powders were loaded into an 80-μmdiameter hole of a stainless steel T301 gasket that was pre-indented to ∼30 μm thickness at ∼18 GPa. With the ruby fluorescence method, the pressure was determined during such an experiment. No pressuretransmitting medium was used to achieve a maximum non-hydrostatic pressure environment. In situ diffraction patterns of the sample were recorded using a two-dimensional imaging plate detector (MAR-3450) for further analysis. The Bragg diffraction rings were converted into a series of one-dimensional intensity versus diffraction angle 2θ patterns using the FIT2D program. It allows for detector calibration and integration of powder diffraction data from 2D detectors to 1D 2θ scans using a CeO2 calibration material from NIST. The Rietveld refinements of powder x-ray diffraction data were carried out using the General Structure Analysis System (GSAS) program as implemented in the EXPGUI package.
2.3. Line-width analysis A polycrystalline sample compressed with non-hydrostatic uniaxial compression in a DAC produces complex stresses and resulting strains in the sample. These stresses can be thought of as a superposition of two types of stresses, macro-differential stress (macro-DS) t and micro-deviatoric stress (micro-DS) ε [23]. There were two different methods for strength of materials under large non-hydrostatic compression: analysis of x-ray diffraction peak broadening and measurement of peak shifts associated with lattice strains. He et al. [24] have reported on the consistency of the two approaches. The diffraction line broadening under non-hydrostatic compression is attributed to two different factors: the reduction in grain size and the presence of micro-strains. The theory proposed earlier for diffraction line broadening from deformed metals was extended for the analysis of high pressure data [25]. The following relation describes the grain size and strain dependencies of diffraction line widths [26]: 2 (2ωhkl cos θhkl )2 = (λ / d )2 + ηhkl sin2 θhkl ,
(1)
where 2ωhkl denotes the full width at half-maximum (FWHM), θhkl is the Bragg angle, λ is the wavelength of the x-ray, and d is the grain size of the crystallites. Meanwhile, ηhkl denotes the microscopic deviatoric strain and depends on Young's modulus E. Once plastic deformation is initialed, macro-DS t and micro-DS ε that the aggregate polycrystalline sample can support are equal to the yield strength Y. The relationship is as follows:
2.2. Theoretical calculations
Y = t= εE ,
(2)
where ε is the average value of ηhkl. The aggregate E can be obtained from the bulk modulus K and shear modulus G using [26]:
In this study, the ab initio simulations based on DFT calculations were performed using the ultrasoft pseudo-potentials plane wave technique, the Cambridge Serial Total Energy Package (CASTEP) code [21]. The exchange-correlation function is described in generalized gradient approximation (GGA) [22]. The elastic modulus and phase stability of TiB2 were obtained by using DFT. Ultrasoft pseudo-potentials were used to characterize the interactions between electrons and the ionic core. The plane wave cut-off energy was chosen as 560 eV, and
E = 9KG /(3K + G )
(3)
3. Results and discussion The AXRD patterns of TiB2 under high pressure up to 42 GPa are 257
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Fig. 2. X-ray diffraction patterns (λ = 0.6199 Å) of TiB2 under various pressures at room temperature. (a) The diffraction patterns of TiB2 upon compression to the highest pressure of 42 GPa. (b) Rietveld refinement patterns of the specimen about 3.1 GPa.
Fig. 3. (a) Experimental and theoretical P–V date for TiB2 up to 42 GPa. (b) Experimental and theoretical cell parameter ratio (a/a0 and c/c0) versus pressure for TiB2. Both a- and c-lattice constant show a gentle decrease upon compression.
K0 = 308(10) GPa of TiB2 with K′0 = 3.4(1) is obtained by fitting to the best Birch-Murnaghan EOS. Apita Aparajita et al. [28] and Ono and Kikegawa [10] have reported the bulk modulus values of TiB2 to be 334 GPa and 256.7 GPa, which are inconsistent with the value of this result. The different bulk modulus results may be attributed to the initial samples in DAC experiment. Moreover, the hydrostatic environment can also lead to the overestimation for the bulk modulus. The calculated P–V data of TiB2 was fitted to the best Birch-Murnaghan EOS, and yielded of K0 = 251(1) GPa with K′0 = 3.8. It is in good agreement with previous results [8,13]. For the purpose of comparison, previously reported values of bulk modulus for TiB2 are tabulated in Table I along with the computational method and the experimental method. In Table I, it is clear that the experimental values are slightly larger than the theoretical calculations. One reason may be that the theoretical values are obtained with an ideal state model. However, the AXRD (hydrostatic and non-hydrostatic) with synchrotron radiation source is the most effective experimental method of measuring the bulk modulus of materials. As shown in Fig. 3a, the experimental points coincide well with the fitting curve. Therefore, the experimental results are reliable and meaningful. Fig. 3b shows the variation of the refined unit cell lattice parameters with pressure. Obviously, the compression behavior of the unit cell axes
exhibited in Fig. 2a. As expected, the hexagonal lattice of TiB2 is stable in the pressure range used in this work. With increasing pressure, it was observed that there is a systematic shift of Bragg peaks towards the higher 2θ angle monotonically, which becomes generally broadened and weakened. Fig. 2b shows one of the diffraction patterns of TiB2 taken at 3.1 GPa pressure together with its least-squares Rietveld refinement using the General Structure Analysis Software (GSAS). Under the pressure of 3.1 GPa, the TiB2 crystals can still maintain its hexagonal lattice structure with the lattice parameters: a=b=3.0274 Å, c=3.2269 Å, which are slightly smaller than those found at zero pressure (a=b=3.0296 Å, c=3.2284 Å), respectively. Fig. 3a shows the pressure evolutions of the unit-cell parameters for TiB2. The bulk modulus was estimated by fitting the pressure versus volume data with the best-order Birch-Murnaghan EOS [27]:
P (V ) =
− 3 ⎡⎛ V ⎞ K0 ⎢ 2 ⎢ ⎝ V0 ⎠ ⎣ ⎜
⎟
7
3
−5 3
V −⎛ ⎞ V ⎝ 0⎠ ⎜
⎟
⎡ V − ⎤ ⎧ 3 ⎥ × 1 + (K 0′ − 4) ⎢ ⎛ ⎞ 4 ⎢ ⎝ V0 ⎠ ⎥ ⎨ ⎦ ⎩ ⎣ ⎜
⎟
2
3
⎤⎫ − 1⎥ , ⎥⎬ ⎦⎭ (4)
where V0 is the unit cell volume at ambient pressure and V is the volume at pressure P. K0 is the isothermal bulk modulus at zero pressure and K 0′ is the pressure derivative of bulk modulus. The bulk modulus of 258
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Table 1 List of the structure parameters (a, b, c), bulk modulus (K0), pressure derivative (K′0) of TiB2, compared with previous results. BM2 denotes a second-order BirchMurnaghan equation of state fixed to 4. BM3 denotes a third-order Birch-Murnaghan equation of state. Method
a, b (Å)
c (Å)
K0 (GPa) (BM2)
K0 (GPa) (BM3)
K 0′ (GPa)
Ref.
Non-hydrostatic compression GGA Hydrostatic compression Ultrasonic Hydrostatic compression GGA GGA GGA LDA
3.0292 3.0298 3.0281(2) 3.0236 3.029(2) 3.015 3.030 3.0291 2.9905
3.2284 3.2206 3.2273(5) 3.2204 3.230(4) 3.222 3.232 3.2195 3.1523
299(3) 247(1) 333(6)
308(10) 250.8(1) 334(7) 237(16) 256.7(7.7) 245
3.4(1) 3.8 3.86 2.0 (2) 3.83(0.28) 3.88
250.6(1) 277.2(3)
3.85(1) 3.84(2)
This work This work [28] [29] [10] [13] [15] [8] [8]
245
Table 2 List of elastic constants Cij (in GPa) of TiB2 from various studies. Shear anisotropy is characterized by the ratio A = C44/C66. Method
C11
C12
C13
C33
C44
C66
A
Ref.
GGA
654(3)
64(1)
97(1)
458(3)
258(1)
295(2)
0.876
GGA LDA GGA GGA GGA Ultrasonic
656(2) 709(3) 655 626 646.5 588
66(1) 71(1) 65 68 76.5 72
98(1) 117(1) 99 102 110 84
461(1) 506(2) 461 444 452 503
259(1) 295(1) 260 240 255 238
295(2) 319(3) 295 279 285 258
0.878 0.925 0.881 0.860 0.895 0.922
This work [8] [8] [12] [13] [15] [29]
Fig. 5. The grain sizes versus pressure for TiB2. Upper-right inset: the (2ωcosθ)2 versus sin2θ plots for TiB2.
(3). Bulk modulus K and shear modulus G can be calculated from the theoretical measurements of cij at high pressure. Fig. 5 shows the grain sizes under different pressures. The instrumental broadening has been considered before Eq. (1) can be used to derive the grain sizes and strains. The (2ωcosθhkl)2 versus sin2θhkl were constructed using the corrected line widths. All experimental points are closer to the relatively straight line that is linearly fitted at each pressure. Examples of such plots are shown in the inset of Fig. 5. Thus, our experimental results are reliable and relatively accurate. Grain size was obtained from the intercept of the straight line drawn through the date points [Eq. (1)]. The grain size decreases steeply during pressurization and remains practically unchanged above 22 GPa. Fig. 6 shows the microscopic deviatoric stress distribution of TiB2 at different pressures, which was derived from Eqs. (1) and (2), and the slope of the line is from the inset of Fig. 5. The microscopic deviatoric strain of TiB2 reached its highest value at a pressure of ∼27 (ε ∼ 0.051) GPa, then fell back for further compression and remained almost the same value as the microscopic deviatoric strain TiB2. This indicates that the microscopic deviatoric stress may reach its limiting value under non-hydrostatic compression and start to yield at around 27 GPa into a plastic deformation. In other words, the yield strength Y of TiB2 was made known at high pressure. It is higher than that of TiB2, which was reported to be ∼1 GPa using x-ray diffraction peak shifts associated with lattice strains methods [17]. Our results demonstrate the consistency of these two methods. As shown in Fig. 6, the yield strength of TiB2 is higher than that of Si3N4 (∼20 GPa) and TaC (∼22 GPa). This may be explained by the hardness of materials. The Vickers hardness of Si3N4 (∼22 GPa), TaC (∼26 GPa), and TiB2 (∼33 GPa) have been intensively reported [30,31]. The inset of Fig. 6 shows the observed three peaks broadening of the initial powder under high pressure up to
Fig. 4. The elastic modulus of titanium diboride at high pressure. The solid triangles, squares, and circles express the aggregate Young's modulus, bulk modulus, and shear modulus, respectively.
is anisotropic. The relative change along the a-axis is about 3.3%, and that along the c-axis is about 4.8%. From 3.1 GPa to 42 GPa for this AXRD experiment, the a-axis is about 5.8% and that along the c-axis is about 8.3% from 0 GPa to 80 GPa for GGA calculation. There is no denying that the elastic constants C11, C22, and C33 represent the incompressibility in a, b, and c directions, respectively. For TiB2, C11 > C33 with C11/C33 = 1.43, its incompressibility in the a direction is much stronger than that in the c direction. The elastic constants Cij of TiB2 along with the values published earlier are summarized in Table II, which are close to those reported by Milman et al. [8]. TiB2 have very high elastic constants consistent with their high hardness. Zener's anisotropy parameter, 2C44/(C11 − C12) = 0.87, is obtained by using firstprinciples calculation in the present work. Fig. 4 shows the elastic modulus of TiB2 as functions of pressures. The Young modulus can also be obtained from the bulk modulus K and shear modulus G using Eq.
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Fig. 6. Differential stress as a function of pressure. Black squares: TiB2; open symbols: other hard materials reported in the literature (blue triangles: TiB2 [17]; red circles: Nano-TaC [30]; pink triangles: Si3N4 [31]). Upper-right inset: full width at half-maximum (FWHM) versus pressure for TiB2. (For interpretation of the references to colour in this figure legend, the reader is referred to the Web version of this article.)
42 GPa. The line widths increase significantly with increasing pressure and remain practically unchanged above 27 GPa. The pressure (∼26 GPa) at which the diffraction peak FWHM tends to be stable is very close to the yield strength (∼27 GPa) for TiB2 under non-hydrostatic compression. 4. Conclusions In summary, in situ AXRD with synchrotron radiation source is performed on TiB2 using a DAC at room temperature. The compression curve of TiB2 can be obtained from highly non-hydrostatic data, and yielded a bulk modulus of K0 = 308 ± 10 GPa with its pressure derivative K′0 = 3.8 ± 1. The strength and plastic properties of TiB2 have been derived by analyzing x-ray diffraction peak broadening. Our experimental results show that the TiB2 sample begins to yield into a plastic deformation at a non-hydrostatic compression of ∼22 GPa and reaches a maximum differential stress of 27 GPa at a given pressure of 42 GPa. Acknowledgements The high-pressure synchrotron AXRD experiments were carried out at beamline 4W2 of the Beijing Synchrotron Radiation Facility (BSRF). This work was supported by the National Natural Science Foundation of China (Grant No. 11604175), and Science and Technology Plan Projects in Qinghai province of China (Grant No: 2014-ZJ-942Q). Appendix A. Supplementary data Supplementary data related to this article can be found at http://dx. doi.org/10.1016/j.jpcs.2018.05.042. References [1] P. Ettmayer, H. Kolaska, W. Lengauer, K. Dreyer, Ti(C,N) cermets-Metallurgy and
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High-pressure strength and compressibility of titanium diboride (TiB2) studied under non-hydrostatic compression
T
Hao Lianga,b, Haihua Chena,∗, Fang Pengb,∗∗, Lingxiao Liua, Xin Lib, Kui Liua, Chuangqi Liuc, Xiaodong Lid a
Foundation Department of Qinghai University, Xining, 810016, PR China Institute of Atomic and Molecular Physics, Sichuan University, Chengdu, 610065, PR China c College of Optoelectronic Technology, Chengdu University of Information Technology, Chengdu, 610225, PR China d Institute of High Energy Physics, Chinese Academy of Sciences, Beijing, 100049, PR China b
A R T I C LE I N FO
A B S T R A C T
Keywords: Titanium diboride Non-hydrostatic pressure Bulk modulus Yield strength
The high-pressure strength and plastic properties of titanium diboride (TiB2) were investigated using synchrotron angle-dispersive x-ray diffraction (AXRD) under non-hydrostatic compression up to 42 GPa in a diamondanvil cell (DAC). The AXRD data yielded a bulk modulus K0 = 308 ± 10 GPa with a pressure derivative K′0 = 3.4 ± 1. The experimental data are discussed and compared to the results of first-principles calculations. The compressibility of TiB2 demonstrates a strongly anisotropic property with increasing pressure. In addition, the microscopic deviatoric stress and grain size (crystallite size) were determined as a function of pressure from the line-width analysis. We can seen that the strength increases while the crystalline size decreases steeply as the pressure is raised from ambient to about 22 GPa. In other words, TiB2 starts to yield a plastic deformation at around 22 GPa, and the yield strength of TiB2 increases with pressure, reaching a value of ∼27 GPa at the highest pressure in our experiment.
1. Introduction Transition metal diboride ceramics (TiB2, CrB2, TaB2, HfB2) are actively investigated because of their useful physical and mechanical characteristics [1]. Titanium diboride (TiB2) materials have received wide attention because of their high melting points (about 3000 °C), high hardness (Vickers hardness = 33.7 GPa and Mohs hardness > 9) and elastic modulus, high resistance to corrosion and wear, and fair chemical inertness in metal melts and excellent thermal shock resistance [2,3]. Due to their outstanding properties, they have been extensively used for high-temperature structural materials, cutting tools and hard coating, and have aroused great interest in basic research and several technological applications [4]. It has been reported that TiB2 has a hexagonal lattice structure (C32) of space group no. 191 (P6/ mmm), and the lattice parameters were a = 3.0292 Å and c = 3.2284 Å. There are three atoms in the unit cell, all of them in special positions: the Hf atom is on the 1a Wyckoff site (0, 0, 0) and the B atoms are on the 2 d Wyckoff site (1/3, 2/3, 1/2) [5]. A hexagonal TiB2 crystal has six different elastic coefficients (C11, C12, C13, C33, C44 and C66), but only five of them are independent since C66 = 1/2 (C11 –
∗
C12) [6]. In previous studies [7–15], the bulk modulus of titanium diboride, which ranges from 193 GPa to 399 GPa, has been intensively investigated using experiments and calculations. Several studies have reported on the bonding mechanism, lattice parameters, and elastic properties of TiB2 using first-principles local-density-functional calculations [16]. Knowledge of the elastic properties, strength, and plastic deformation of TiB2 ceramic materials is a requisite for understanding its stability and for its applications under extreme static and dynamic mechanical stresses and high temperature. In 2006, TiB2 was studied by Amulele et al. [17] in a DAC in order to determine its hydrostatic equation of state (EOS) and investigate its strength. The measurements were conducted up to 60 GPa by using the radial x-ray diffraction technique and analyzed using the lattice strain theory [18–20]. Unfortunately, there are relatively few studies devoted to direct experimental measurements on the strength and plastic deformation behavior of titanium diboride by analyzing the x-ray diffraction peak broadening under non-hydrostatic compression. In this study, the EOS, high-pressure strength and plastic properties of a polycrystalline titanium diboride sample were investigated using a diamond anvil cell (DAC) by synchrotron angle-dispersive x-ray
Corresponding author. Foundation Department of Qinghai University, Xining, 810016, PR China. Corresponding author. Institute of Atomic and Molecular Physics, Sichuan University, Chengdu, 610065, PR China. E-mail addresses: [email protected] (H. Chen), [email protected] (F. Peng).
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https://doi.org/10.1016/j.jpcs.2018.05.042 Received 15 December 2017; Received in revised form 25 May 2018; Accepted 26 May 2018 Available online 26 May 2018 0022-3697/ © 2018 Published by Elsevier Ltd.
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Fig. 1. (a) X-ray diffraction patterns (λ = 1.5404 Å) of the initial powder under normal pressure and room temperature. Upper-right inset: scanning electron microscope (SEM) images of starting powder. (b) Schematic crystal structures of TiB2.
9 × 9 × 2 Monkhorst–Pack k-point sampling was used for all cases. The tolerances for geometry optimization were set as the difference of total energy within 5 × 10−6 5 eV/atom, maximum ionic Hellmann-Feynman force within 0.01 eV/Å, maximum ionic displacement within 5 × 10−4 Å, and maximum stress within 0.02 GPa. Convergence tests revealed that the parameters described above were sufficient to lead to a well-converged total energy, which are carefully tested in the text. The bulk modulus at ambient pressure (K0) and its pressure derivative (K 0′) were determined by fitting the numerical data to the isothermal Birch-Murnaghan EOS.
diffraction (AXRD) under uniaxial compression up to 42 GPa. For comparison with previous and experimental results, the elastic properties of TiB2 were calculated using first-principles density functional theory (DFT). 2. Material and methods 2.1. Experimental procedures Polycrystalline titanium diboride powder (99.99% purity, 0.5–1 μm grain size) was purchased from the Alfa Aesar Co., Ltd., Shanghai China. The initial powder was characterized by x-ray diffraction (XRD; model DX-2500, Dandong, China) using a Cu Kα radiation source with λ = 1.5404 Å to check the crystal structure, purity and morphology. Fig. 1a shows the power XRD pattern and scanning electron microscope (SEM) images of the starting specimen. As shown in Fig. 1b, TiB2 has a hexagonal structure. An in situ high pressure AXRD experiment was carried out using a symmetric-type DAC with a culet size of 300 μm, which was interfaced with angle-dispersive synchrotron radiation at the 4W2 beamline of the Beijing Synchrotron Radiation Facility (BSRF, China) at room temperature. TiB2 powders were loaded into an 80-μmdiameter hole of a stainless steel T301 gasket that was pre-indented to ∼30 μm thickness at ∼18 GPa. With the ruby fluorescence method, the pressure was determined during such an experiment. No pressuretransmitting medium was used to achieve a maximum non-hydrostatic pressure environment. In situ diffraction patterns of the sample were recorded using a two-dimensional imaging plate detector (MAR-3450) for further analysis. The Bragg diffraction rings were converted into a series of one-dimensional intensity versus diffraction angle 2θ patterns using the FIT2D program. It allows for detector calibration and integration of powder diffraction data from 2D detectors to 1D 2θ scans using a CeO2 calibration material from NIST. The Rietveld refinements of powder x-ray diffraction data were carried out using the General Structure Analysis System (GSAS) program as implemented in the EXPGUI package.
2.3. Line-width analysis A polycrystalline sample compressed with non-hydrostatic uniaxial compression in a DAC produces complex stresses and resulting strains in the sample. These stresses can be thought of as a superposition of two types of stresses, macro-differential stress (macro-DS) t and micro-deviatoric stress (micro-DS) ε [23]. There were two different methods for strength of materials under large non-hydrostatic compression: analysis of x-ray diffraction peak broadening and measurement of peak shifts associated with lattice strains. He et al. [24] have reported on the consistency of the two approaches. The diffraction line broadening under non-hydrostatic compression is attributed to two different factors: the reduction in grain size and the presence of micro-strains. The theory proposed earlier for diffraction line broadening from deformed metals was extended for the analysis of high pressure data [25]. The following relation describes the grain size and strain dependencies of diffraction line widths [26]: 2 (2ωhkl cos θhkl )2 = (λ / d )2 + ηhkl sin2 θhkl ,
(1)
where 2ωhkl denotes the full width at half-maximum (FWHM), θhkl is the Bragg angle, λ is the wavelength of the x-ray, and d is the grain size of the crystallites. Meanwhile, ηhkl denotes the microscopic deviatoric strain and depends on Young's modulus E. Once plastic deformation is initialed, macro-DS t and micro-DS ε that the aggregate polycrystalline sample can support are equal to the yield strength Y. The relationship is as follows:
2.2. Theoretical calculations
Y = t= εE ,
(2)
where ε is the average value of ηhkl. The aggregate E can be obtained from the bulk modulus K and shear modulus G using [26]:
In this study, the ab initio simulations based on DFT calculations were performed using the ultrasoft pseudo-potentials plane wave technique, the Cambridge Serial Total Energy Package (CASTEP) code [21]. The exchange-correlation function is described in generalized gradient approximation (GGA) [22]. The elastic modulus and phase stability of TiB2 were obtained by using DFT. Ultrasoft pseudo-potentials were used to characterize the interactions between electrons and the ionic core. The plane wave cut-off energy was chosen as 560 eV, and
E = 9KG /(3K + G )
(3)
3. Results and discussion The AXRD patterns of TiB2 under high pressure up to 42 GPa are 257
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Fig. 2. X-ray diffraction patterns (λ = 0.6199 Å) of TiB2 under various pressures at room temperature. (a) The diffraction patterns of TiB2 upon compression to the highest pressure of 42 GPa. (b) Rietveld refinement patterns of the specimen about 3.1 GPa.
Fig. 3. (a) Experimental and theoretical P–V date for TiB2 up to 42 GPa. (b) Experimental and theoretical cell parameter ratio (a/a0 and c/c0) versus pressure for TiB2. Both a- and c-lattice constant show a gentle decrease upon compression.
K0 = 308(10) GPa of TiB2 with K′0 = 3.4(1) is obtained by fitting to the best Birch-Murnaghan EOS. Apita Aparajita et al. [28] and Ono and Kikegawa [10] have reported the bulk modulus values of TiB2 to be 334 GPa and 256.7 GPa, which are inconsistent with the value of this result. The different bulk modulus results may be attributed to the initial samples in DAC experiment. Moreover, the hydrostatic environment can also lead to the overestimation for the bulk modulus. The calculated P–V data of TiB2 was fitted to the best Birch-Murnaghan EOS, and yielded of K0 = 251(1) GPa with K′0 = 3.8. It is in good agreement with previous results [8,13]. For the purpose of comparison, previously reported values of bulk modulus for TiB2 are tabulated in Table I along with the computational method and the experimental method. In Table I, it is clear that the experimental values are slightly larger than the theoretical calculations. One reason may be that the theoretical values are obtained with an ideal state model. However, the AXRD (hydrostatic and non-hydrostatic) with synchrotron radiation source is the most effective experimental method of measuring the bulk modulus of materials. As shown in Fig. 3a, the experimental points coincide well with the fitting curve. Therefore, the experimental results are reliable and meaningful. Fig. 3b shows the variation of the refined unit cell lattice parameters with pressure. Obviously, the compression behavior of the unit cell axes
exhibited in Fig. 2a. As expected, the hexagonal lattice of TiB2 is stable in the pressure range used in this work. With increasing pressure, it was observed that there is a systematic shift of Bragg peaks towards the higher 2θ angle monotonically, which becomes generally broadened and weakened. Fig. 2b shows one of the diffraction patterns of TiB2 taken at 3.1 GPa pressure together with its least-squares Rietveld refinement using the General Structure Analysis Software (GSAS). Under the pressure of 3.1 GPa, the TiB2 crystals can still maintain its hexagonal lattice structure with the lattice parameters: a=b=3.0274 Å, c=3.2269 Å, which are slightly smaller than those found at zero pressure (a=b=3.0296 Å, c=3.2284 Å), respectively. Fig. 3a shows the pressure evolutions of the unit-cell parameters for TiB2. The bulk modulus was estimated by fitting the pressure versus volume data with the best-order Birch-Murnaghan EOS [27]:
P (V ) =
− 3 ⎡⎛ V ⎞ K0 ⎢ 2 ⎢ ⎝ V0 ⎠ ⎣ ⎜
⎟
7
3
−5 3
V −⎛ ⎞ V ⎝ 0⎠ ⎜
⎟
⎡ V − ⎤ ⎧ 3 ⎥ × 1 + (K 0′ − 4) ⎢ ⎛ ⎞ 4 ⎢ ⎝ V0 ⎠ ⎥ ⎨ ⎦ ⎩ ⎣ ⎜
⎟
2
3
⎤⎫ − 1⎥ , ⎥⎬ ⎦⎭ (4)
where V0 is the unit cell volume at ambient pressure and V is the volume at pressure P. K0 is the isothermal bulk modulus at zero pressure and K 0′ is the pressure derivative of bulk modulus. The bulk modulus of 258
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Table 1 List of the structure parameters (a, b, c), bulk modulus (K0), pressure derivative (K′0) of TiB2, compared with previous results. BM2 denotes a second-order BirchMurnaghan equation of state fixed to 4. BM3 denotes a third-order Birch-Murnaghan equation of state. Method
a, b (Å)
c (Å)
K0 (GPa) (BM2)
K0 (GPa) (BM3)
K 0′ (GPa)
Ref.
Non-hydrostatic compression GGA Hydrostatic compression Ultrasonic Hydrostatic compression GGA GGA GGA LDA
3.0292 3.0298 3.0281(2) 3.0236 3.029(2) 3.015 3.030 3.0291 2.9905
3.2284 3.2206 3.2273(5) 3.2204 3.230(4) 3.222 3.232 3.2195 3.1523
299(3) 247(1) 333(6)
308(10) 250.8(1) 334(7) 237(16) 256.7(7.7) 245
3.4(1) 3.8 3.86 2.0 (2) 3.83(0.28) 3.88
250.6(1) 277.2(3)
3.85(1) 3.84(2)
This work This work [28] [29] [10] [13] [15] [8] [8]
245
Table 2 List of elastic constants Cij (in GPa) of TiB2 from various studies. Shear anisotropy is characterized by the ratio A = C44/C66. Method
C11
C12
C13
C33
C44
C66
A
Ref.
GGA
654(3)
64(1)
97(1)
458(3)
258(1)
295(2)
0.876
GGA LDA GGA GGA GGA Ultrasonic
656(2) 709(3) 655 626 646.5 588
66(1) 71(1) 65 68 76.5 72
98(1) 117(1) 99 102 110 84
461(1) 506(2) 461 444 452 503
259(1) 295(1) 260 240 255 238
295(2) 319(3) 295 279 285 258
0.878 0.925 0.881 0.860 0.895 0.922
This work [8] [8] [12] [13] [15] [29]
Fig. 5. The grain sizes versus pressure for TiB2. Upper-right inset: the (2ωcosθ)2 versus sin2θ plots for TiB2.
(3). Bulk modulus K and shear modulus G can be calculated from the theoretical measurements of cij at high pressure. Fig. 5 shows the grain sizes under different pressures. The instrumental broadening has been considered before Eq. (1) can be used to derive the grain sizes and strains. The (2ωcosθhkl)2 versus sin2θhkl were constructed using the corrected line widths. All experimental points are closer to the relatively straight line that is linearly fitted at each pressure. Examples of such plots are shown in the inset of Fig. 5. Thus, our experimental results are reliable and relatively accurate. Grain size was obtained from the intercept of the straight line drawn through the date points [Eq. (1)]. The grain size decreases steeply during pressurization and remains practically unchanged above 22 GPa. Fig. 6 shows the microscopic deviatoric stress distribution of TiB2 at different pressures, which was derived from Eqs. (1) and (2), and the slope of the line is from the inset of Fig. 5. The microscopic deviatoric strain of TiB2 reached its highest value at a pressure of ∼27 (ε ∼ 0.051) GPa, then fell back for further compression and remained almost the same value as the microscopic deviatoric strain TiB2. This indicates that the microscopic deviatoric stress may reach its limiting value under non-hydrostatic compression and start to yield at around 27 GPa into a plastic deformation. In other words, the yield strength Y of TiB2 was made known at high pressure. It is higher than that of TiB2, which was reported to be ∼1 GPa using x-ray diffraction peak shifts associated with lattice strains methods [17]. Our results demonstrate the consistency of these two methods. As shown in Fig. 6, the yield strength of TiB2 is higher than that of Si3N4 (∼20 GPa) and TaC (∼22 GPa). This may be explained by the hardness of materials. The Vickers hardness of Si3N4 (∼22 GPa), TaC (∼26 GPa), and TiB2 (∼33 GPa) have been intensively reported [30,31]. The inset of Fig. 6 shows the observed three peaks broadening of the initial powder under high pressure up to
Fig. 4. The elastic modulus of titanium diboride at high pressure. The solid triangles, squares, and circles express the aggregate Young's modulus, bulk modulus, and shear modulus, respectively.
is anisotropic. The relative change along the a-axis is about 3.3%, and that along the c-axis is about 4.8%. From 3.1 GPa to 42 GPa for this AXRD experiment, the a-axis is about 5.8% and that along the c-axis is about 8.3% from 0 GPa to 80 GPa for GGA calculation. There is no denying that the elastic constants C11, C22, and C33 represent the incompressibility in a, b, and c directions, respectively. For TiB2, C11 > C33 with C11/C33 = 1.43, its incompressibility in the a direction is much stronger than that in the c direction. The elastic constants Cij of TiB2 along with the values published earlier are summarized in Table II, which are close to those reported by Milman et al. [8]. TiB2 have very high elastic constants consistent with their high hardness. Zener's anisotropy parameter, 2C44/(C11 − C12) = 0.87, is obtained by using firstprinciples calculation in the present work. Fig. 4 shows the elastic modulus of TiB2 as functions of pressures. The Young modulus can also be obtained from the bulk modulus K and shear modulus G using Eq.
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Fig. 6. Differential stress as a function of pressure. Black squares: TiB2; open symbols: other hard materials reported in the literature (blue triangles: TiB2 [17]; red circles: Nano-TaC [30]; pink triangles: Si3N4 [31]). Upper-right inset: full width at half-maximum (FWHM) versus pressure for TiB2. (For interpretation of the references to colour in this figure legend, the reader is referred to the Web version of this article.)
42 GPa. The line widths increase significantly with increasing pressure and remain practically unchanged above 27 GPa. The pressure (∼26 GPa) at which the diffraction peak FWHM tends to be stable is very close to the yield strength (∼27 GPa) for TiB2 under non-hydrostatic compression. 4. Conclusions In summary, in situ AXRD with synchrotron radiation source is performed on TiB2 using a DAC at room temperature. The compression curve of TiB2 can be obtained from highly non-hydrostatic data, and yielded a bulk modulus of K0 = 308 ± 10 GPa with its pressure derivative K′0 = 3.8 ± 1. The strength and plastic properties of TiB2 have been derived by analyzing x-ray diffraction peak broadening. Our experimental results show that the TiB2 sample begins to yield into a plastic deformation at a non-hydrostatic compression of ∼22 GPa and reaches a maximum differential stress of 27 GPa at a given pressure of 42 GPa. Acknowledgements The high-pressure synchrotron AXRD experiments were carried out at beamline 4W2 of the Beijing Synchrotron Radiation Facility (BSRF). This work was supported by the National Natural Science Foundation of China (Grant No. 11604175), and Science and Technology Plan Projects in Qinghai province of China (Grant No: 2014-ZJ-942Q). Appendix A. Supplementary data Supplementary data related to this article can be found at http://dx. doi.org/10.1016/j.jpcs.2018.05.042. References [1] P. Ettmayer, H. Kolaska, W. Lengauer, K. Dreyer, Ti(C,N) cermets-Metallurgy and
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